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operations research an introduction

Questions and Answers of Operations Research An Introduction

Post-Optimal Analysis for Cases Affecting Both Optimality and Feasibility. Suppose that you are given the following simultaneous changes in the Reddy Mikks model:The revenue per ton of exterior and
Show that the 100% feasibility rule in Problem 3-79 (Chapter 3) is based on the condition aOptimum inverse b a Original right@hand side vector b Ú 0
The Ozark Farm has 20,000 broilers that are fed for 8 weeks before being marketed.The weekly feed per broiler varies according to the following schedule:Week 1 2 3 4 5 6 7 8 lb/broiler .26 .48 .75
Consider the Reddy Mikks model of Example 2.1-1. Its optimal tableau is given in Example 3.3-1. If the daily availabilities of raw materials M1 and M2 are increased to 35 and 10 tons, respectively,
Suppose that TOYCO wants to change the capacities of the three operations according to the following cases:(a) °460 500 400¢ (b) °500 400 600¢ (c) °300 800 200¢ (d) °450 700 350¢Use
In the TOYCO model listed at the start of Section 4.5, would it be more advantageous to assign the 20-minute excess capacity of operation 3 to operation 2 instead of operation 1?
The LP model of Problem 4-38(d) has no bounded solution. Show how this condition is detected by the generalized simplex procedure.
The LP model of Problem 4-38(c) has no feasible solution. Show how this condition is detected by the generalized simplex procedure.
Solve the following LP in three different ways (use TORA for convenience). Which method appears to be the most efficient computationally?Minimize z = 6x1 + 7x2 + 3x3 + 5x4 subject to 5x1 + 6x2 - 3x3
Using the artificial constraint procedure introduced in Problem 4-37, solve the following problems by the dual simplex method. In each case, indicate whether the resulting solution is feasible,
Dual Simplex with Artificial Constraints. Consider the following problem:Maximize z = 2x1 - x2 + x3 subject to 2x1 + 3x2 - 5x3 Ú 4 -x1 + 9x2 - x3 Ú 3 4x1 + 6x2 + 3x3 … 8 x1, x2, x3 Ú 0 The
Generate the dual simplex iterations for the following problems (using TORA for convenience), and trace the path of the algorithm on the graphical solution space.(a) Minimize z = 2x1 + 3x2 subject to
Consider the solution space in Figure 4.3, where it is desired to find the optimum extreme point that uses the dual simplex method to minimize z = 2x1 + x2. The optimal solution occurs at point F =
The company estimates that for each part that is not produced (per the optimum solution), an across-the-board 20%reduction in machining time can be realized through process improvements. Would these
Consider the optimal solution of JoShop in Problem
JoShop uses lathes and drill presses to produce four types of machine parts, PP1, PP2, PP3, and PP4. The following table summarizes the pertinent data:Machining time in minutes per unit of Machine
In Example 4.3-2, suppose that TOYCO is studying the possibility of introducing a fourth toy: fire trucks. The assembly does not make use of operation 1. Its unit assembly times on operations 2 and 3
In Example 4.3-2, suppose that for toy trains the per-unit time of operation 2 can be reduced from 3 minutes to at most 1.3 minutes. By how much must the per-unit time of operation 1 be reduced to
BagCo produces leather jackets and handbags. A jacket requires 8 m2 of leather, and a handbag only 2 m2. The labor requirements for the two products are 12 and 5 hours, respectively. The current
NWAC Electronics manufactures four types of simple cables for a defense contractor.Each cable must go through four sequential operations: splicing, soldering, sleeving, and inspection. The following
In Example 4.3-1, compute the change in the optimal revenue in each of the following cases (use TORA output to obtain the feasibility ranges):(a) The constraint for raw material M1 (resource 1) is
Show that Method 1 in Section 4.2.3 for determining the optimal dual values is actually based on the Formula 2 in Section 4.2.4.
Consider the following LP:Maximize z = 2x1 + 4x2 + 4x3 - 3x4 subject to x1 + x2 + x3 = 4 x1 + 4x2 + x4 = 8 x1, x2, x3, x4 Ú 0 Use the dual problem to show that the basic solution (x1, x2) is not
The following is the optimal tableau for a maximization LP model with three (…)constraints and all nonnegative variables. The variables x3, x4, and x5 are the slacks associated with the three
Consider the following LP model:Maximize z = 5x1 + 2x2 + 3x3 subject to x1 + 5x2 + 2x3 … b1 x1 - 5x2 - 6x3 … b2 x1, x2, x3 Ú 0 The following optimal tableau corresponds to specific values of b1
Consider the following LP model:Maximize z = 5x1 + 12x2 + 4x3 subject to x1 + 2x2 + x3 + x4 = 5 2x1 - x2 + 3x3 = 1 x1, x2, x3, x4 Ú 0(a) Identify the best solution from among the following basic
Consider the following LP model:Minimize z = 2x1 + x2 subject to 3x1 + x2 - x3 = 3 4x1 + 3x2 - x4 = 6 x1 + 2x2 + x5 = 3 x1, x2, x3, x4, x5 Ú 0 Compute the entire simplex tableau associated with the
Consider the following LP model:Maximize z = 3x1 + 2x2 + 5x3 subject to x1 + 2x2 + x3 + x4 = 30 3x1 + 2x3 + x5 = 60 x1 + 4x2 + x6 = 20 x2, x2, x3, x4, x5, x6 Ú 0 Check the optimality and feasibility
Consider the following LP model:Maximize z = 4x1 + 14x2 subject to 2x1 + 7x2 + x3 = 21 7x1 + 2x2 + x4 = 21 x1, x2, x3, x4 Ú 0 Check the optimality and feasibility of each of the following basic
Generate the first simplex iteration of Example 4.2-1 (you may use TORA’s Iterations 1 M@method with M = 100 for convenience), then use Formulas 1 and 2 to verify all the elements of the resulting
In Problem 4-17(a), let y1 and y2 be the dual variables. Determine whether the following pairs of primal–dual solutions are optimal:*(a) (x1 = 3, x2 = 1; y1 = 4, y2 = 1)(b) (x1 = 4, x2 = 1; y1 = 1,
Estimate a range for the optimal objective value for the following LPs:*(a) Minimize z = 5x1 + 2x2 subject to x1 - x2 Ú 3 2x1 + 3x2 Ú 5 x1, x2 Ú 0(b) Maximize z = x1 + 5x2 + 3x3 subject to x1 +
Consider the following set of inequalities:2x1 + 3x2 … 12-3x1 + 2x2 … -4 3x1 - 5x2 … 2 x1 unrestricted x2 Ú 0 A feasible solution can be found by augmenting the trivial objective function,
Consider the following LP:Maximize z = x1 + 5x2 + 3x3 subject to x1 + 2x2 + x3 = 3 2x1 - x2 = 4 x1, x2, x3 Ú 0 The starting solution consists of x3 in the first constraint and an artificial x4 in
Consider the following LP:Maximize z = 2x1 + 4x2 + 4x3 - 3x4 subject to x1 + x2 + x3 = 4 x1 + 4x2 + x4 = 8 x1, x2, x3, x4 Ú 0 Using x3 and x4 as starting variables, the optimal tableau is given as
Consider the following LP:Minimize z = 4x1 + x2 subject to 3x1 + x2 = 30 4x1 + 3x2 Ú 60 x1 + 2x2 … 40 x1, x2 Ú 0 The starting solution consists of artificial x4 and x5 for the first and second
Consider the following LP:Maximize z = 5x1 + 2x2 + 3x3 subject to x1 + 5x2 + 2x3 = 15 x1 - 5x2 - 6x3 … 20 x1, x2, x3 Ú 0 Given that the artificial variable x4 and the slack variable x5 form the
Solve the dual of the following problem, and then find its optimal solution from the solution of the dual. Does the solution of the dual offer computational advantages over solving the primal
Find the optimal value of the objective function for the following problem by inspecting only its dual. (Do not solve the dual by the simplex method.)Minimize z = 10x1 + 4x2 + 5x3 subject to 5x1 -
Repeat Problem 4-8 for the last tableau of Example 3.4-2.
Consider the optimal tableau of Example 3.3-1.*(a) Identify the optimal inverse matrix.(b) Show that the right-hand side equals the inverse multiplied by the original right-hand side vector of the
Consider the following matrices:a = £1 4 2 5 3 6, p1 = a 10 20b, p2 = £10 20 30V1 = 111, 222, V2 = 1 -2, -4, -62 In each of the following cases, indicate whether the given matrix operation is
True or False?(a) The dual of the dual problem yields the original primal.(b) If the primal constraint is originally in equation form, the corresponding dual variable is necessarily unrestricted.(c)
Consider Example 4.1-1. The application of the simplex method to the primal requires the use of an artificial variable in the second constraint of the standard primal to secure a starting basic
Write the dual for each of the following primal problems:(a) Maximize z = 66x1 - 22x2 subject to-x1 + x2 … -2 2x1 + 3x2 … 5 x1, x2 Ú 0(b) Minimize z = 6x1 + 3x2 subject to 6x1 - 3x2 + x3 Ú 25
In Example 4.1-3, show that even if the sense of optimization in the primal is changed to minimization, an unrestricted primal variable always corresponds to an equality dual constraint.
In Example 4.1-2, derive the associated dual problem given that the primal problem is augmented with a third constraint, 3x1 + x2 = 4.
In Example 4.1-1, derive the associated dual problem if the sense of optimization in the primal problem is changed to minimization.
Consider Problem 2-76 (Chapter 2).(a) Which of the specification constraints impacts the optimum solution adversely?(b) Is it economical for the company to purchase ore 1 at $100/ton. Explain in
Consider Problem 2-49, (Chapter 2).(a) Suppose that the manufacturer can purchase additional units of raw material A at$12 per unit. Would it be advisable to do so?(b) Would you recommend that the
Consider Problem 2-48, (Chapter 2). Suppose that any additional capacity of machines 1 and 2 can be acquired only by using overtime. What is the maximum cost per hour the company should be willing to
Consider Problem 2-47, (Chapter 2). Relate the dual prices to the unit production costs of the model.
Consider Problem 2-45 (Chapter 2). Use the dual price to decide if it is advisable for the gambler to bet an additional $400.
Consider Problem 2-44, (Chapter 2). Use the dual price to determine if it is worthwhile for the executive to invest more money in the plans.
Consider Problem 2-43, (Chapter 2). Use the dual prices to determine the rate of return associated with each year.
Consider Problem 2-42 (Chapter 2).(a) Give an economic interpretation of the dual prices of the model.(b) Show how the dual price associated with the upper bound on borrowed money at the beginning of
Consider Problem 2-41 (Chapter 2).(a) Use the dual prices to determine the overall return on investment.(b) If you wish to spend $2000 on pleasure at the end of year 1, how would this affect the
Consider Problem 2-40 (Chapter 2). Use the dual price to decide if it is worthwhile to increase the funding for year 4.22
The 100% Optimality Rule. A rule similar to the 100% feasibility rule outlined in Problem 3-79, can also be developed for testing the effect of simultaneously changing all cj to cj + dj, j = 1, 2,c,
Dean’s Furniture Company assembles regular and deluxe kitchen cabinets from precut lumber. The regular cabinets are painted white, and the deluxe are varnished. Both painting and varnishing are
Popeye Canning is contracted to receive daily 50,000 lb of ripe tomatoes at 7 cents per pound, from which it produces canned tomato juice, tomato sauce, and tomato paste. The canned products are
Electra produces four types of electric motors, each on a separate assembly line. The respective capacities of the lines are 500, 500, 800, and 750 motors per day. Type 1 motor uses 8 units of a
The Bank of Elkins is allocating a maximum of $200,000 for personal and car loans during the next month. The bank charges 14% for personal loans and 12% for car loans.Both types of loans are repaid
Baba Furniture Company employs four carpenters for 10 days to assemble tables and chairs. It takes 2 person-hours to assemble a table and half a person-hour to assemble a chair. Customers usually buy
B&K grocery store sells three types of soft drinks: the brand names A1 Cola, A2 Cola, and the cheaper store brand BK Cola. The price per can for A1, A2, and BK are 80, 70, and 60 cents, respectively.
In the TOYCO model, determine if the current solution will change in each of the following cases:21(i) z = x1 + x2 + 4x3(ii) z = 4x1 + 6x2 + x3(iii) z = 6x1 + 3x2 + 9x3
Consider the problem Maximize z = x1 + x2 subject to 2x1 + x2 … 6 x1 + 2x2 … 6 x1 + x2 Ú 0(a) Show that the optimal basic solution includes both x1 and x2 and that the feasibility ranges for the
The 100% feasibility rule. A simplified rule based on the individual changes D1, D2, . . . , and Dm in the right-hand side of the constraints can be used to test whether or not simultaneous changes
HiDec produces two models of electronic gadgets that use resistors, capacitors, and chips.The following table summarizes the data of the situation:Unit resource requirements Resource Model 1 (units)
The Gutchi Company manufactures purses, shaving bags, and backpacks. The construction of the three products requires leather and synthetics, with leather being the limiting raw material. The
An assembly line consisting of three consecutive workstations produces two radio models: DiGi-1 and DiGi-2. The following table provides the assembly times for the three workstations.Minutes per unit
ChemLabs uses raw materials I and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively.One unit of solution A
The Burroughs Garment Company manufactures men’s shirts and women’s blouses for Walmark Discount Stores. Walmark will accept all the production supplied by Burroughs.The production process
Show & Sell can advertise its products on local radio and television (TV), or in newspapers.The advertising budget is limited to $10,000 a month. Each minute of advertising on radio costs $15 and
The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical, such as woodworking, word
A company that operates 10 hrs a day manufactures three products on three processes.The following table summarizes the data of the problem:Minutes per unit Product Process 1 Process 2 Process 3 Unit
A company produces three products, A, B, and C. The sales volume for A is at least 50%of the total sales of all three products. However, the company cannot sell more than 80 units of A per day. The
Consider the TOYCO model.(a) Suppose that any additional time for operation 1 beyond its current capacity of 430 mins per day must be done on an overtime basis at $50 an hour. The hourly cost
In the TOYCO model, suppose that the changes D1, D2, and D3 are made simultaneously in the three operations.20(a) If the availabilities of operations 1, 2, and 3 are changed to 440, 490, and 400
In Problem 3-64:(a) Determine the optimality range for the unit revenue ratio of the two types of hats that will keep the current optimum unchanged.(b) Using the information in (a), will the optimal
In the Reddy Mikks model of Example 2.2-1:(a) Determine the range for the ratio of the unit revenue of exterior paint to the unit revenue of interior paint.(b) If the revenue per ton of exterior
Consider Problem 3-63.(a) Determine the optimality condition for cA cB that will keep the optimum unchanged.(b) Determine the optimality ranges for cA and cB, assuming that the other coefficient is
A company produces two products, A and B. The unit revenues are $2 and $3, respectively.Two raw materials, M1 and M2, used in the manufacture of the two products have daily availabilities of 8 and 18
Consider the LP model Maximize z = 3x1 + 2x1 + 3x3 subject to 2x1 + x2 + x3 … 4 3x1 + 4x2 + 2x3 Ú 16 x1, x2, x3 Ú 0 Use hand computations to show that the optimal solution can include an
Toolco produces three types of tools, T1, T2, and T3. The tools use two raw materials, M1 and M2, according to the data in the following table:Number of units of raw materials per tool Raw material
In some ill-constructed LP models, the solution space may be unbounded even though the problem may have a bounded objective value. Such an occurrence points to possible irregularities in the
Consider the LP:Maximize z = 20x1 + 5x2 + x3 subject to 3x1 + 5x2 - 5x3 … 50 x1 … 10 x1 + 3x2 - 4x3 … 20 x1, x2, x3 Ú 0(a) By inspecting the constraints, determine the direction (x1, x2, or
TORA Experiment. Solve Example 3.5-3 using TORA’s Iterations option and show that even though the solution starts with x1 as the entering variable (per the optimality condition), the simplex
For the following LP, show that the optimal solution is degenerate and that none of the alternative solutions are corner points. You may use TORA for convenience.Maximize z = 3x1 + x2 subject to x1 +
Solve the following LP:Maximize z = 2x1 - x2 + 3x3 subject to x1 - x2 + 5x3 … 5 2x1 - x2 + 3x3 … 20 x1, x2, x3 Ú 0 From the optimal tableau, show that all the alternative optima are not corner
For the following LP, identify three alternative optimal basic solutions, and then write a general expression for all the nonbasic alternative optima comprising these three basic solutions.Maximize z
TORA Experiment. Consider the following LP (authored by E.M. Beale to demonstrate cycling):Maximize z = 34 x1 - 20x2 + 12 x3 - 6x4 subject to 14 x1 - 8x2 - x3 + 9x4 … 0 12 x1 - 12x2 - 12 x3 + 3x4
TORA experiment. Consider the LP in Problem 3-52.(a) Use TORA to generate the simplex iterations. How many iterations are needed to reach the optimum?(b) Interchange constraints (1) and (3) and
Consider the following LP:Maximize z = 3x1 + 2x2 subject to 4x1 - x2 … 4 4x1 + 3x2 … 6 4x1 + x2 … 4 x1, x2 Ú 0 (a) Show that the associated simplex iterations are temporarily degenerate (you
Consider the graphical solution space in Figure 3.18. Suppose that the simplex iterations start at A and that the optimum solution occurs at D. Further, assume that the objective function is defined
Consider the LP model Minimize z = 2x1 - 4x2 + 3x3 subject to 5x1 - 6x2 + 2x3 Ú 5-x1 + 3x2 + 5x3 Ú 8 2x1 + 5x2 - 4x3 … 4 x1, x2, x3 Ú 0 Show how the inequalities can be modified to a set of
Consider the following LP:Maximize z = 3x1 + 2x2 + 3x3 subject to 2x1 + x2 + x3 … 2 3x1 + 4x2 + 2x3 Ú 8 x1, x2, x3 Ú 0 The optimal simplex tableau at the end of Phase I is Basic x1 x2 x3 x4 x5 R
Consider the following problem:Maximize z = 3x1 + 2x2 + 3x3 subject to 2x1 + x2 + x3 = 4 x1 + 3x2 + x3 = 12 3x1 + 4x2 + 2x3 = 16 x1, x2, x3 Ú 0(a) Show that Phase I terminates with two zero
Consider the following problem:Maximize z = 2x1 + 2x2 + 4x3 subject to 2x1 + x2 + x3 … 2 3x1 + 4x2 + 2x3 Ú 8 x1, x2, x3 Ú 0(a) Show that Phase I will terminate with an artificial basic variable
Write Phase I for the following problem, and then solve (with TORA for convenience) to show that the problem has no feasible solution.Minimize z = 2x1 + 5x2 subject to 3x1 + 2x2 Ú 12 2x1 + x2 … 4

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